Posted on by Kalkicode
Code Mathematics

# Average of cubes of first n natural numbers

Calculating the average of cubes of the first n natural numbers involves finding the sum of the cubes of the numbers from 1 to n and then dividing that sum by n. This is a simple mathematical problem that can be solved using a formula that leverages the properties of summation.

## Problem Statement

Given a positive integer n, the problem is to find the average of the cubes of the first n natural numbers.

## Idea to Solve the Problem

The sum of the cubes of the first n natural numbers can be represented by the sum of the cubes of consecutive integers, which is a well-known formula: 1³ + 2³ + 3³ + ... + n³. The formula for the sum of cubes is given by:

Sum = 1³ + 2³ + 3³ + ... + n³ = (n * (n + 1) / 2)²

The average of these cubes can be calculated by dividing the sum by n.

## Pseudocode

``````function findAverageOfCube(n)
if n <= 0
return

sum_of_cubes = (n * (n + 1) / 2)^2
average = sum_of_cubes / n

print average``````

## Algorithm Explanation

1. The `findAverageOfCube` function takes a positive integer `n` as input.

2. If `n` is less than or equal to 0, the function returns immediately to handle invalid inputs.

3. Calculate the sum of the cubes of the first n natural numbers using the formula `(n * (n + 1) / 2)^2`.

4. Calculate the average by dividing the sum of cubes by n.

5. Print the calculated average.

## Program Solution

``````// C Program
// Average of cubes of first n natural numbers
#include <stdio.h>

void findAverageOfCube(int n)
{
if (n <= 0)
{
return;
}
// Calculate cube of n natural number
// Formula = (n×(n+1)²) / 4
double average = (n * (n + 1) * (n + 1)) / 4.0;
// Display calculated result
printf("\n %lf", average);
}
int main()
{
int n = 6;
/*
n = 6
---------------------------
1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (440)
----------------------------
440
-- = 73.5
6

*/
findAverageOfCube(n);
return 0;
}``````

#### Output

`` 73.500000``
``````// Java program for
// Average of cubes of first n natural numbers
public class Average
{
public void findAverageOfCube(int n)
{
if (n <= 0)
{
return;
}
// Calculate cube of n natural number
// Formula = (n×(n+1)²) / 4
double average = (n * (n + 1) * (n + 1)) / 4.0;
// Display calculated result
System.out.print("\n " + average);
}
public static void main(String[] args)
{
int n = 6;
/*
n = 6
---------------------------
1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (440)
----------------------------
440
-- = 73.5
6

*/
}
}``````

#### Output

`` 73.5``
``````// Include header file
#include <iostream>

using namespace std;
// C++ program for
// Average of cubes of first n natural numbers
class Average
{
public: void findAverageOfCube(int n)
{
if (n <= 0)
{
return;
}
// Calculate cube of n natural number
// Formula = (n×(n+1)²) / 4
double average = (n *(n + 1) *(n + 1)) / 4.0;
// Display calculated result
cout << "\n " << average;
}
};
int main()
{
int n = 6;
/*
n = 6
---------------------------
1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (440)
----------------------------
440
-- = 73.5
6
*/
return 0;
}``````

#### Output

`` 73.5``
``````// Include namespace system
using System;
// Csharp program for
// Average of cubes of first n natural numbers
public class Average
{
public void findAverageOfCube(int n)
{
if (n <= 0)
{
return;
}
// Calculate cube of n natural number
// Formula = (n×(n+1)²) / 4
double average = (n * (n + 1) * (n + 1)) / 4.0;
// Display calculated result
Console.Write("\n " + average);
}
public static void Main(String[] args)
{
int n = 6;
/*
n = 6
---------------------------
1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (440)
----------------------------
440
-- = 73.5
6
*/
}
}``````

#### Output

`` 73.5``
``````package main
import "fmt"
// Go program for
// Average of cubes of first n natural numbers
type Average struct {}
func getAverage() * Average {
var me *Average = &Average {}
return me
}
func(this Average) findAverageOfCube(n int) {
if n <= 0 {
return
}
// Calculate cube of n natural number
// Formula = (n×(n+1)²) / 4
var average float64 = float64(n * (n + 1) * (n + 1)) / 4.0
// Display calculated result
fmt.Print("\n ", average)
}
func main() {
var task * Average = getAverage()
var n int = 6
/*
n = 6
---------------------------
1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (440)
----------------------------
440
-- = 73.5
6
*/
}``````

#### Output

`` 73.5``
``````<?php
// Php program for
// Average of cubes of first n natural numbers
class Average
{
public	function findAverageOfCube(\$n)
{
if (\$n <= 0)
{
return;
}
// Calculate cube of n natural number
// Formula = (n×(n+1)²) / 4
\$average = (\$n * (\$n + 1) * (\$n + 1)) / 4.0;
// Display calculated result
echo("\n ".\$average);
}
}

function main()
{
\$n = 6;
/*
n = 6
---------------------------
1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (440)
----------------------------
440
-- = 73.5
6
*/
}
main();``````

#### Output

`` 73.5``
``````// Node JS program for
// Average of cubes of first n natural numbers
class Average
{
findAverageOfCube(n)
{
if (n <= 0)
{
return;
}
// Calculate cube of n natural number
// Formula = (n×(n+1)²) / 4
var average = (n * (n + 1) * (n + 1)) / 4.0;
// Display calculated result
process.stdout.write("\n " + average);
}
}

function main()
{
var n = 6;
/*
n = 6
---------------------------
1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (440)
----------------------------
440
-- = 73.5
6
*/
}
main();``````

#### Output

`` 73.5``
``````#  Python 3 program for
#  Average of cubes of first n natural numbers
class Average :
def findAverageOfCube(self, n) :
if (n <= 0) :
return

#  Calculate cube of n natural number
#  Formula = (n×(n+1)²) / 4
average = (n * (n + 1) * (n + 1)) / 4.0
#  Display calculated result
print("\n ", average, end = "")

def main() :
n = 6
#    n = 6
#    ---------------------------
#    1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (440)
#    ----------------------------
#    440
#    -- = 73.5
#     6

if __name__ == "__main__": main()``````

#### Output

``  73.5``
``````#  Ruby program for
#  Average of cubes of first n natural numbers
class Average
def findAverageOfCube(n)
if (n <= 0)
return
end

#  Calculate cube of n natural number
#  Formula = (n×(n+1)²) / 4
average = (n * (n + 1) * (n + 1)) / 4.0
#  Display calculated result
print("\n ", average)
end

end

def main()
n = 6
#    n = 6
#    ---------------------------
#    1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (440)
#    ----------------------------
#    440
#    -- = 73.5
#     6
end

main()``````

#### Output

`` 73.5``
``````// Scala program for
// Average of cubes of first n natural numbers
class Average()
{
def findAverageOfCube(n: Int): Unit = {
if (n <= 0)
{
return;
}
// Calculate cube of n natural number
// Formula = (n×(n+1)²) / 4
var average: Double = (n * (n + 1) * (n + 1)) / 4.0;
// Display calculated result
print("\n " + average);
}
}
object Main
{
def main(args: Array[String]): Unit = {
var task: Average = new Average();
var n: Int = 6;
/*
n = 6
---------------------------
1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (440)
----------------------------
440
-- = 73.5
6
*/
}
}``````

#### Output

`` 73.5``
``````// Swift 4 program for
// Average of cubes of first n natural numbers
class Average
{
func findAverageOfCube(_ n: Int)
{
if (n <= 0)
{
return;
}
// Calculate cube of n natural number
// Formula = (n×(n+1)²) / 4
let average: Double = Double(n * (n + 1) * (n + 1)) / 4.0;
// Display calculated result
print("\n ", average, terminator: "");
}
}
func main()
{
let n: Int = 6;
/*
n = 6
---------------------------
1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (440)
----------------------------
440
-- = 73.5
6
*/
}
main();``````

#### Output

``  73.5``
``````// Kotlin program for
// Average of cubes of first n natural numbers
class Average
{
fun findAverageOfCube(n: Int): Unit
{
if (n <= 0)
{
return;
}
// Calculate cube of n natural number
// Formula = (n×(n+1)²) / 4
val average: Double = (n * (n + 1) * (n + 1)) / 4.0;
// Display calculated result
print("\n " + average);
}
}
fun main(args: Array < String > ): Unit
{
val n: Int = 6;
/*
n = 6
---------------------------
1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (440)
----------------------------
440
-- = 73.5
6
*/
}``````

#### Output

`` 73.5``

## Time Complexity

The time complexity of this algorithm is constant (O(1)) because the calculations involve basic arithmetic operations that do not depend on the input size.

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